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# Linear Transformations

**Keywords:**Middle-division, quantum mechanics, linear algebra, eigenvectors

### Highlights of the activity

- This small group activity is designed to help students understand the geometry behind transformation matrices.
- Students are given a list of vectors to draw in different colors. Each group is assigned one of 12 matrices. The students are then instructed to operate on the vectors with the matrix, and observe the changes in the vectors.
- The compare and contrast wrap-up discussion focuses on the changes caused by the different matrices and the class as a whole proposes hypotheses about the geometric meaning of the determinant of the matrix. Furthermore, this activity also brought up the topic of the vectors that are unchanged by the matrix, led the students to identify those vectors as the eigenvectors of the matrix.

### Reasons to spend class time on the activity

As students enter their junior-year physics classes, they view basic objects in linear algebra as algebraic rather than geometric. To communicate the physical meaning of an eigenvector, we begin with the very geometric understanding of vectors that students remember from their early courses. This activity provides students with a geometric understanding of eigenvalues, and eigenvectors as well as a quick review of basic matrix manipulations.